Explicit solutions of Genz test integrals

Abstract

A collection of test integrals introduced by Genz (1984) has remained popular to this day for assessing the robustness of high-dimensional numerical integration algorithms. However, the explicit solutions to these integrals do not appear to be readily available in the existing literature: typically the true values of the test integrals are simply approximated using “overkill” numerical solutions. In this paper, analytic solutions are presented for the Genz test integrals ∫0 ⋯ 1 ∫0 cos ( 2𝜋 𝑤1 + Σ𝑑 𝑖=1 𝑐𝑖𝑥𝑖 ) d𝑥𝑑 ⋯ d𝑥1 = 2𝑑 cos ( 2𝜋 𝑤1 + 12 Σ𝑑 𝑖=1 𝑐𝑖 )Π𝑑 𝑖=1 sin( 𝑐𝑖 2 ) 𝑐𝑖 , 1 ∫0 ⋯ 1 ∫0 Π𝑑 𝑖=1 1 𝑐−2 𝑖 + (𝑥𝑖 − 𝑤𝑖)2 d𝑥𝑑 ⋯ d𝑥1 = Π𝑑 𝑖=1 𝑐𝑖 ( ar ct an(𝑐𝑖𝑤𝑖) + ar ct an(𝑐𝑖 − 𝑐𝑖𝑤𝑖) ) , 1 ∫0 ⋯ 1 ∫0 ( 1 + Σ𝑑 𝑖=1 𝑐𝑖𝑥𝑖 )−(𝑑+1) d𝑥𝑑 ⋯ d𝑥1 = 1 𝑑! Π𝑑𝑖=1 𝑐𝑖 Σ u⊆{𝑐1 ,…,𝑐𝑑 } (−1)#u 1 + Σ 𝑖∈u 𝑖 , 1 ∫0 ⋯ 1 ∫0 exp ( − Σ𝑑 𝑖=1 𝑐2 𝑖 (𝑥𝑖 − 𝑤𝑖)2 ) d𝑥𝑑 ⋯ d𝑥1 = 𝜋𝑑∕2 2𝑑 Π𝑑 𝑖=1 er f (𝑐𝑖𝑤𝑖) + er f (𝑐𝑖 − 𝑐𝑖𝑤𝑖) 𝑐𝑖 , 1 ∫0 ⋯ 1 ∫0 exp ( − Σ𝑑 𝑖=1 𝑐𝑖|𝑥𝑖 − 𝑤𝑖| ) d𝑥𝑑 ⋯ d𝑥1 = Π𝑑 𝑖=1 exp(𝑐𝑖𝑤𝑖 − 𝑐𝑖) − exp(−𝑐𝑖𝑤𝑖) 𝑐𝑖 , 𝑤1 ∫0 𝑤2 ∫0 1 ∫0 ⋯ 1 ∫0 exp (Σ𝑑 𝑖=1 𝑐𝑖𝑥𝑖 ) d𝑥𝑑 ⋯ d𝑥3 d𝑥2 d𝑥1 = Π2 𝑖=1 (exp(𝑐𝑖𝑤𝑖) − 1) Π𝑑𝑖 =3(exp(𝑐𝑖) − 1) Π𝑑𝑖 =1 𝑐𝑖 , where 𝑑 ∈ Z+, 0 < 𝑤𝑖 < 1, and 𝑐𝑖 ∈ R+ for all 𝑖 ∈ {1,…, 𝑑}.